The distance between two points $\(P(6,0) \& Q(0,8)\)$, which are situated at the two opposite ends of the circle C , will be the diameter of the circle.

So, the length of the diameter PQ is $\(\sqrt{(6-0)^2+(0-8)^2}=\sqrt{36+64}=\sqrt{100}=10\)$

$\( Distance between points\) $\(\left(x_1, y_1\right) \&\left(x_2, y_2\right)\)$ is $\(\left.\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} \)$

Thus, the radius of the circle is $\(\frac{1}{2}($ Diameter $)=\frac{1}{2} \times 10=5\)$

Hence the answer is (C).